The relationship determination between the Bayesian networks nodes based on the intuitionistic fuzzy set

Abstract

For the Bayesian network (BN) structure learning, the key problem is to determine the relationship between the BN nodes. In this paper, the scheme of group decision making (GDM) based on the intuitionistic fuzzy set for the relationship determination between the BN nodes is proposed. Firstly, the alternative relationships between the BN nodes are analyzed. The relationship determination problem is transformed into the GDM problem. Furthermore, the specific GDM scheme is proposed to determine the relationship. Finally, the proposed scheme is applied to establish the model for the thickening process of gold hydrometallurgy. For the different conditions of group expert knowledge including the consistent and inconsistent information, the process of GDM is shown, and the aggregation results of different aggregation operators and the influence of hesitancy degree are analyzed. We can conclude that the expert who owns bigger membership degree and less hesitancy degree plays the most important role in the process of decision making.

Keywords

Bayesian network relationship determination group decision making expert knowledge intuitionistic fuzzy set inconsistent information

1. Introduction

Bayesian network (BN) is a powerful tool to represent the conditional probability relationships among the variables in the form of a directed acyclic graph, which has been used in the various fields, such as information fusion [24, 32], fault diagnosis [11, 26] and prediction [17, 21]. For the establishment of BN, two aspects need to be considered: parameter and structure. For the structure learning, the key problem is to determine the relationships among the variables. There are two ways to determine the relationships including expert knowledge methods [19, 26] and data-oriented methods [3, 6, 30]. The structure learning of BNs from data is an NP-hard problem [10], especially when the data are scarce and the related variables are huge in the problem domain. At the same time, the collected data are always imprecise, which are influenced by the noise interference and the precision of sensors. Therefore, when the researched domain includes available expert knowledge, it is an effective way to determine the relationship between the BN nodes using the expert knowledge. Especially in the real industrial environment, the expert knowledge is easy to be collected from the experts or operators. When integrating the expert knowledge to determine some relationships among the variables for the data-oriented methods, the search space of alternative structures is reduced greatly, and the best BN structure can be obtained as soon as possible. Therefore, the introduction of expert knowledge is beneficial for boosting the performance of structure learning methods.

Before using the expert knowledge, the first problem is how to express the expert knowledge in a suitable way. The traits of expert knowledge are imprecise and vague in nature. At the same time, the decision makers have limited ability to process a large amount of related information. Therefore, the evaluations on the relationships among the nodes are often uncertain and subjective. The experts may be unable to discriminate the causal relationship between the variables explicitly. In such cases, it is difficult to express the expert knowledge using the crisp numbers because of the inherent ambiguous, vague and subjective characteristics of human judgments. In order to solve the problem, the intuitionistic fuzzy set (IFS) theory proposed by Atanassov [2] is a useful tool to deal with vagueness, ambiguity and hesitation. Compared with the FS [31], except the membership function, the IFS is associated with the non-membership function and the hesitancy function. That is to say, in the FS, the sum of membership degree and non-membership degree is equal to one. However, in the IFS, the sum of membership degree, non-membership degree and hesitancy degree is equal to one. The hesitancy degree is introduced to represent the ignorance, which represents the lack of knowledge for the membership degree. The superiority of IFS has been proofed broadly in the aspects of representing and integrating the expert knowledge. In many aspects, such as evaluation, negotiation and decision making, the hesitancy degree plays an important role when the decision makers may not express their opinions on the alternative solutions accurately. The IFS has been widely applied in many fields involved in the decision making problems [5, 14, 23], pattern recognition [9, 18], and so on. When determining the relationship between the BN nodes, the objective is to determine the most suitable relationship among all the feasible alternative relationships. It can be regarded as a decision-making problem. To reduce the information bias from the single decision maker, in this paper, we consider the group experts to express opinions and make decisions. Different experts may hold different opinions on the same problem. Therefore, it is not only necessary to integrate the opinions of group experts but also to consider the integration problem of inconsistent information.

In this paper, first of all, by analyzing the alternative relationships between the BN nodes and collecting the expert knowledge in the form of intuitionistic fuzzy values (IFVs), the problem of relationship determination between the BN nodes is transformed into the group decision making (GDM) problem based on the IFS. Furthermore, the GDM scheme is proposed to determine the relationship between the BN nodes. Finally, the proposed scheme is applied to establish the model for the thickening process of gold hydrometallurgy. For the different conditions, the influences of different aggregation operators on the aggregation results and the role of hesitancy degree in the decision-making process are compared and analyzed. Two operators, the IFWA operator and the aggregation operator from the Dempster-Shafer (D-S) theory, are chosen to aggregate the opinions from the group experts respectively. The consistent and inconsistent information are analyzed respectively. The aggregation results are compared by changing the value of hesitancy degree in the IFVs. By analyzing the different aggregation results, the role of hesitancy degree is obtained in the process of decision making for the inconsistent information.

The novelty of this paper reflects three aspects. At first, this paper provides a new angle and scheme to determine the relationship between the BN nodes. The specific GDM scheme based on the IFS firstly is proposed to solve this problem. Furthermore, when collecting the expert knowledge in the form of IFVs, the subjective knowledge degree information is introduced to construct the expert knowledge base. This way is more reasonable to express the expert knowledge, which combines the subjective and objective factors. Finally, the proposed scheme is applied to establish the model for the thickening process of gold hydrometallurgy. By comparing the aggregation results of different conditions, the role of hesitancy degree is firstly obtained.

The remaining sections of this paper are organized as follows. The Section 2 discusses some related work. The Section 3 discusses some related concepts. In the Section 4, the problem of relationship determination between the BN nodes is analyzed, which is transformed into the GDM problem. The corresponding decision scheme is proposed. In the Section 5, the proposed scheme is applied to establish the BN structure for the thickening process of gold hydrometallurgy. Some conditions are analyzed to demonstrate the validity of proposed scheme for the consistent and inconsistent information aggregation problems. The aggregation results of different aggregation operators are shown and compared, and the role of hesitancy degree is discussed. Finally, the conclusions are presented in the Section 6.

2. Related work

Many research results [1, 7, 11, 20] have been proposed by integrating the expert knowledge into the BN structure learning process. In the paper [26], the BN structure is built based on the diagnostic criteria and the domain experts. In the paper [7], the proposed method needs to request the direct probabilistic relationships between the variables from the experts. In the paper [20], an interactive approach is proposed to integrate domain/expert knowledge at the different stages of learning BN structure. However, the above papers don’t explain that how to use the expert knowledge, which only show the final expression form of expert knowledge in the BN structure learning. Therefore, in this paper, we mainly analyze the specific process of using the expert knowledge when determining the relationship between the BN nodes and solve the related problems. The paper [25] proposes a methodology by transforming expert knowledge or final GDM statements into a set of qualitative statements and probability inequality constraints for inference in a BN. However, to the best of our knowledge, few researches [1, 7, 19, 20, 26] focus on the relationship determination problem between the BN nodes in the intuitionistic fuzzy environment. In addition, in the related researches of IFS, few results [5, 14, 23] discuss the role of hesitancy degree in the decision making process, especially for the inconsistent information aggregation. Therefore, the relationship determination problem between the BN nodes based on the IFS and the role of hesitancy degree in the decision-making process will be considered in this paper.

3. Preliminary

3.1 Bayesian network (BN) [22]

BN is a kind of effective method for the probabilistic representing and reasoning in the artificial intelligence. The form of BN is a directed acyclic graph. The nodes in the BN represent the variables; the arcs represent the direct causal relationships between the linked nodes. The nodes and the directed arcs constitute the BN structure. The conditional probability tables are assigned to the nodes to specify how strongly the linked nodes influence each other, which constitute the BN parameters.

3.2 Intuitionistic fuzzy set (IFS) [2]

An IFS $A$ in a finite set, $X$ is an object having the following form:

$\displaystyle A=\{\langle x,\mu_{A}(x),\nu_{A}(x)\rangle|x\in X\}$ (1)

where $\mu_{A}(x):X\to[0,1]$ and $\nu_{A}(x):X\to[0,1]$ such that $\mu_{A}(x)+\nu_{A}(x)\leqslant 1$ for $\forall x\in X$ , and they denote the degree of membership and the degree of non-membership of $x\in A$ , respectively.

Another parameter of IFS is $\pi_{A}(x)=1-\mu_{A}(x)-\nu_{A}(x)$ , known as intuitionistic fuzzy index (or hesitation margin) of $x\in A$ , and it expresses a lack of knowledge of whether $x$ belongs to $A$ or not. It is clear that $\forall x\in X,0\leqslant\pi_{A}(x)\leqslant 1$ . Obviously, when $\mu_{A}(x)=1-\nu_{A}(x)$ , the concept of FS is covered.

3.3 Intuitionistic fuzzy weighted averaging (IFWA) [29]

Let $A_{i}=\langle\mu_{A_{i}},\nu_{A_{i}}\rangle(i=1,2,\ldots,n)$ be a collection of IFVs, $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})^{T}$ be the aggregation-associated weight vector such that $\omega_{i}\in[0,1]$ and $\sum_{i=1}^{n}{\omega_{i}}=1$ . Then we have the following aggregation operator. The IFWA operator:

$\displaystyle\textit{IFWA}(A_{1},A_{2},\ldots,A_{n})=\omega_{1}A_{1}\oplus% \omega_{2}A_{2}\oplus\ldots\oplus\omega_{n}A_{n}=\left\langle 1-\prod\limits_{% i=1}^{n}(1-\mu_{A_{i}})^{\omega_{i}},\prod\limits_{i=1}^{n}\nu_{A_{i}}^{\omega% _{i}}\right\rangle$ (2)

3.4 Aggregation operator from the D-S theory [28]

Let $B_{1}=\langle\mu_{1},\nu_{1}\rangle$ and $B_{2}=\langle\mu_{2},\nu_{2}\rangle$ be two IFVs. A combination operation on $B_{1}$ and $B_{2}$ , denoted by $B_{1}\oplus B_{2}$ , is defined by:

$\displaystyle B_{1}\oplus B_{2}=\langle(\mu_{1}(1-\nu_{2})+\mu_{2}\pi_{1})/(1-% \mu_{1}\nu_{2}-\mu_{2}\nu_{1}),$

(3) $\displaystyle\quad(\nu_{1}(1-\mu_{2})+\nu_{2}\pi_{1})/(1-\mu_{1}\nu_{2}-\mu_{2% }\nu_{1})\rangle$

The hesitancy degree of $B_{1}\oplus B_{2}$ is:

$\displaystyle\pi_{B_{1}\oplus B_{2}}=(\pi_{1}\pi_{2})/(1-\mu_{1}\nu_{2}-\mu_{2% }\nu_{1})$ (4)

It is obvious that $B_{1}\oplus B_{2}$ is also an IFV. The Dempster’s rule is applied to combine the IFVs. Above combination operation is from the D-S theory.
3.5 Knowledge measure [15]

A mapping $K:\text{IFS}(X)\to[0,1]$ is called a knowledge measure on $\text{IFS}(X)$ , if $K$ has the following properties:

(i)
$K(A)=1$ if $A$ is a crisp set;
(ii)
$K(A)=0$ if $\pi_{A}(x_{i})=1\forall x_{i}\in X(i=1,2,\ldots,n)$ ;
(iii)
$K(A)\geqslant K(B)$ if $A$ is less fuzzy than $B$ , i.e., $\mu_{A}(x_{i})\leqslant\mu_{B}(x_{i})$ and $\nu_{A}(x_{i})\geqslant\nu_{B}(x_{i})$ for $\mu_{B}(x_{i})\leqslant\nu_{B}(x_{i})\forall x_{i}\in X$ or $\mu_{A}(x_{i})\geqslant\mu_{B}(x_{i})$ and $\nu_{A}(x_{i})\leqslant\nu_{B}(x_{i})$ for $\mu_{B}(x_{i})\geqslant\nu_{B}(x_{i})\linebreak\forall x_{i}\in X$ ;
(iv)
$K(A^{c})=K(A)$ , where $A=<\mu_{A}(x_{i}),\nu_{A}(x_{i})>$ , $A^{c}=<\nu_{A}(x_{i}),\mu_{A}(x_{i})>$ .

For $\forall A\in\text{IFS}(X)$ , the knowledge measure model is as follow [15]:

$\displaystyle K_{\text{AIFS}}(A)=1-[1/(2n)]\sum\limits_{i=1}^{n}{(1-\left|{\mu% _{A}(x_{i})-\nu_{A}(x_{i})}\right|)}(1+\pi_{A}(x_{i}))$ (5)

The knowledge measure is related with the information provided by the experts [15], which represents how much information contains in the provided opinions. When the hesitation degrees of IFVs are the same, the bigger the membership degree of IFV is, and the bigger the knowledge measure is. When the membership degrees of IFVs are the same, the bigger the hesitation degree of IFV is, and the smaller the knowledge measure is.
4. GDM scheme based on the IFS

4.1 Problem formulation

For the BN structure learning, the key problem is to determine the relationships among the variables. It is an effective way to determine the relationship using the expert knowledge. Especially when the data are scarce, and the related variables are huge in the researched domain, the expert knowledge can reduce the uncertainty of models. In the following part, we will analyze the relationship between the BN nodes.

After determining the related variables as the nodes of BN in the problem domain, the expert knowledge on the relationships between the BN nodes need to be collected. Taking the relationship between two nodes $X$ and $Y$ as the example, it includes three alternative relationships: $X\to Y$ ; $X\leftarrow Y$ ; $X\neq Y$ . $X\to Y$ represents that $X$ points to $Y$ , that is to say, $X$ is the parent node (cause node) and $Y$ is the children node (result node). In the same way, $X\leftarrow Y$ represents that $Y$ points to $X$ , that is to say, $Y$ is the parent node (cause node) and $X$ is the children node (result node). $X\neq Y$ represents that the two nodes have no relationship. For the three alternative relationships between two nodes, the group experts need to express their opinions and form the initial knowledge base. The expression form of expert knowledge is the IFS. By the decision-making method, the most suitable relationship between the nodes is determined. Therefore, the problem of relationship determination between the BN nodes is transformed into the GDM problem based on the IFS. In this paper, $E=\{E_{1},E_{2},\ldots,E_{k}\},(1\leqslant i\leqslant k)$ represent the experts. $A=\{a_{1},a_{2},\ldots,a_{n}\},(1\leqslant j\leqslant n)$ represent the alternative relationships. Therefore, the above problem in the intuitionistic fuzzy environment can be concisely expressed in the matrix format as follow:

$\displaystyle D=(\langle\mu_{ij},\nu_{ij}\rangle)_{k\times n}=\left({{\begin{% array}[]{*{20}c}{\langle\mu_{11},\nu_{11}\rangle}&{\langle\mu_{12},\nu_{12}% \rangle}&\ldots&{\langle\mu_{1n},\nu_{1n}\rangle}\\ {\langle\mu_{21},\nu_{21}\rangle}&{\langle\mu_{22},\nu_{22}\rangle}&\ldots&{% \langle\mu_{2n},\nu_{2n}\rangle}\\ \vdots&\vdots&\ddots&\vdots\\ {\langle\mu_{k1},\nu_{k1}\rangle}&{\langle\mu_{k2},\nu_{k2}\rangle}&\ldots&{% \langle\mu_{kn},\nu_{kn}\rangle}\\ \end{array}}}\right)$ (6)

where $D$ represents the intuitionistic fuzzy decision matrix. The column represents the alternative relationships and the row represents the expert. $\langle\mu_{ij},\nu_{ij}\rangle$ represents that the evaluation of $i$ th expert on $j$ th alternative relationship. Because in this paper there are three alternative relationships between the BN nodes, we can obtain that $n=3$ .

4.2 Proposed GDM scheme

In this section, for the proposed problem in the Section 4.1, the GDM scheme based on the IFS is shown in the following Algorithm. The steps of proposed GDM scheme are shown to determine the most appropriate relationship between the BN nodes.

Step 1.
Obtain the knowledge base from the group experts for each alternative relationship between the BN nodes in the intuitionistic fuzzy environment. The form is shown in the matrix Eq. (6).

In the intuitionistic fuzzy environment, three elements need to be determined, membership degree $\mu_{A}(x)$ , non-membership degree $\nu_{A}(x)$ and hesitancy degree $\pi_{A}(x)$ . The sum of three elements is equal to one, that is to say, $\mu_{A}(x)+\nu_{A}(x)+\pi_{A}(x)=1$ . Therefore, after two elements are obtained, the third element can be calculated by the simple subtraction. In general, when collecting the expert knowledge in the intuitionistic fuzzy environment, the membership degree and the non-membership degree are obtained from the group experts directly. These are regarded as the objective information for the relationship determination. In this paper, we change the above general expert knowledge collecting method by combining the subjective and objective information to construct the knowledge base. Firstly, we introduce the concept of knowledge degree. Knowledge degree refers to the evaluations on the experts, which are based on the understanding on the researched domain and the academic level of every expert. It is expressed by a number belonging to $[{0,1}]$ . Only can the opinions of experts who own the bigger (more than 0.6) knowledge degree play the important role in the decision-making process. In the proposed expert knowledge collecting method, the membership degree and the hesitancy degree are obtained firstly. Specifically, the membership degree still is obtained from the expert directly. The hesitancy degree represents a lack of knowledge, which is determined by the knowledge degree. The determination of hesitancy degree can follow the guideline that the bigger the knowledge degree is, the smaller the hesitancy degree is. The change range of hesitancy degree $\pi_{A}(x)$ is $[{0,1-\mu_{A}(x)}]$ . Based on the above qualitative guideline, the hesitancy degree can be obtained. Therefore, after obtaining the membership degree and the hesitancy degree, the non-membership degree can be calculated by $\nu_{A}(x)=1-\mu_{A}(x)-\pi_{A}(x)$ . Finally, the knowledge base from the group experts can be established. The knowledge degree is a kind of subjective information, and the hesitancy degree is obtained based on the knowledge degree. Therefore, the established knowledge base includes the subjective and objective information simultaneously.
Step 2.
Calculate the weight of each expert.

In the decision-making process, it is an important problem that how to determine the weights of experts, which has high influence on the final decision results. $W=\{\omega_{1},\omega_{2},\ldots,\omega_{k}\}$ are the weights of experts, which represent the importance degrees of experts. The acquisition of weights mainly includes two ways: the subjective way and the objective way. The former ones are determined by evaluating the importance of experts based on their knowledge for the specific background; the latter ones are determined by the obtained information from the experts. The knowledge measure and the entropy method are important approaches to determine the objective weights, which measures the degree of fuzziness and the amount of contained knowledge from the obtained information [4, 15]. In this paper, the knowledge measure is used to determine the weight of each expert. Based on the Section 3.5, the knowledge measure of each expert can be obtained by Eq. (5), represented as $K_{i}(1\leqslant i\leqslant k)$ . It can be used to determine the weight of each expert by the following form.

$\displaystyle\omega_{K_{i}}=K_{i}\left/\sum\limits_{i=1}^{k}K_{i}(1\leqslant i% \leqslant k)\right.$ (7)

The determination of weight based on the knowledge measure is a typical objective method. Because the subjective and objective information are all included in the collected knowledge base, the determination of weight includes the subjective and objective information indirectly.
Step 3.
Aggregate the information from the group experts to obtain the comprehensive evaluation on the alternative relationships between the BN nodes.

In the intuitionistic fuzzy framework, a large number of aggregation operators have been proposed to aggregate the information, such as the IFWA, the intuitionistic fuzzy weighted geometric (IFWG), and so on. In addition, the aggregation operator from the D-S theory has attracted more and more attention [8, 12, 13, 28]. The choice of operators will influence the aggregation results of IFVs. In this step, to compare and analyze the aggregation results of different aggregation operators for the different conditions, we choose two operators, the IFWA operator and the aggregation operator from the D-S theory, to aggregate the opinions from the group experts respectively. The IFWA is one of the most common aggregation operators in the intuitionistic fuzzy environment, which is shown in Eq. (2). The aggregation operator from the D-S theory is shown in Eqs (3.4) and (4). Based on the research results in the papers [8, 28], when using the aggregation operator from the D-S theory, the IFVs need be handled by the discounting operation. In this paper, we use the weights of experts as the discount coefficients. The specific aggregation process can refer to the related research results in the papers [8, 28], which is omitted here. The aggregation results will be compared and analyzed in the simulation analysis section.
Step 4.
Calculate the relative closeness coefficient of every alternative relationship.

Based on the technique for order preference by similarity to ideal solution (TOPSIS) method [16], the distances from the alternative relationships to the ideal solutions should be calculated. The ideal solutions include the positive-ideal solution and negative-ideal solution, which are represented as $\langle 1,0\rangle$ and $\langle 0,1\rangle$ . The distance is calculated by the following equation [27]

$\displaystyle d=1/2(\left|{\mu_{A}(x)-\mu_{B}(x)}\right|+\left|{\nu_{A}(x)-\nu% _{B}(x)}\right|{}+\left|{\mu_{B}(x)+\nu_{B}(x)-\mu_{A}(x)-\nu_{A}(x)}\right|)$ (8)

The relative closeness coefficient $\xi_{j}(1\leqslant j\leqslant n)$ can be calculated by the following equation [5]

$\displaystyle\xi_{j}=d_{j-}/(d_{j-}+d_{j+})(1\leqslant j\leqslant n)$ (9)

where $d_{j+}$ represents the distance from the $j$ th alternative relationship to the positive-ideal solution. $d_{j-}$ represents the distance from the $j$ th alternative relationship to the negative-ideal solution.
Step 5.
Rank all relative closeness coefficients and select the most suitable relationship between the BN nodes.

The most suitable relationship between the BN nodes is the one which owns the highest value of all the relative closeness coefficients.

Algorithm The GDM scheme based on the IFS

Require: Intuitionistic fuzzy decision matrix D

Ensure: The most suitable relationship between the BN nodes

1: $i=1$

2: repeat

3: for $\{\langle\mu_{i1},\nu_{i1}\rangle\ldots\langle\mu_{in},\nu_{in}\rangle\}\in D$ do

4: Obtain the knowledge measure of $i$ th expert based on Eq. (5)

5: Calculate the weight of $i$ th expert based on Eq. (7)

6: $i=i+1$

7: until $i>k$

8: Aggregate the information from the group experts based on Eqs (2) or (3.4)

9: Calculate the relative closeness coefficient of every alternative relationship based on Eqs (8) and (9)

10: Rank all relative closeness coefficients

11: Choose the relationship which owns the maximum relative closeness coefficient as the most suitable relationship between the BN nodes

The role of expert in the process of relationship determination between the BN nodes is shown in the Fig. 1. To solve the target problem, we need to analyze the process variables and extract the related variables as the BN nodes. The establishment of model structure is transformed into the relationship determination among the nodes. The group experts are invited to express their opinions on the relationship between two nodes until all the relationships among the nodes can be determined. When the experts express their opinions, it involves four problems: how to express the expert knowledge, how to determine the weight of each expert, how to aggregate the opinions of group experts and how to make a decision on the aggregation results. The proposed algorithm provides a scheme to solve the above problem.

Figure 1.
The role of expert in the process of relationship determination between the BN nodes.

5. Simulation analysis

Algorithm The GDM scheme based on the IFS
Require: Intuitionistic fuzzy decision matrix D
Ensure: The most suitable relationship between the BN nodes
1: $i=1$
2: repeat
3: for $\{\langle\mu_{i1},\nu_{i1}\rangle\ldots\langle\mu_{in},\nu_{in}\rangle\}\in D$ do
4: Obtain the knowledge measure of $i$ th expert based on Eq. (5)
5: Calculate the weight of $i$ th expert based on Eq. (7)
6: $i=i+1$
7: until $i>k$
8: Aggregate the information from the group experts based on Eqs (2) or (3.4)
9: Calculate the relative closeness coefficient of every alternative relationship based on Eqs (8) and (9)
10: Rank all relative closeness coefficients
11: Choose the relationship which owns the maximum relative closeness coefficient as the most suitable relationship between the BN nodes

In this section, the proposed GDM scheme is applied to establish the model for the thickening process of gold hydrometallurgy. We will discuss the following two problems by the way of simulation analysis.

The first problem: in the decision-making process, the opinions from the group experts may be consistent or inconsistent. Therefore, how to combine the information of different conditions becomes a challenging problem in the decision-making process. In the following simulation analysis, we will carry out the proposed scheme in steps. At the same time, the aggregation results of two aggregation operators, the IFWA operator and the aggregation operator from the D-S theory, are shown and compared for the different conditions, which provide the basis for the choice of aggregation operators for the different conditions. The Examples 5.1–5.4 will discuss this problem.

The second problem: as far as we know, few researches discuss the role of hesitancy degree in the decision-making process for the inconsistent information. Therefore, in the following part, we will analyze the role of hesitancy degree by changing the value of hesitancy degree in the IFVs and observing the change of aggregation results. The Examples 5.5–5.6 will discuss this problem.

5.1 The thickening process of gold hydrometallurgy

The gold hydrometallurgical process consists of the following main sub-processes: flotation, concentration, leaching, washing and cementation. Before the cyanide leaching, the slurry needs to be concentrated to get the high solid. In this paper, we call this process as the thickening process which is a key process to guarantee efficiency of the following cyanide leaching. The simplified schematic diagram of thickening process is shown in the Fig. 2.

Figure 2.

The simplified schematic diagram of thickening process.

In the thickening process, the main equipment includes thickener, pressure filter and buffer slots. The slurry pumps and the valves regulate the slurry rate among them. In this paper, we use the two common abnormities of thickening process as the research objects. The two abnormities are described as the following forms: (1) The underflow concentration of thickener is too high; (2) Buffer slot 1 under the thickener is empty. The specific description about the abnormities in the thickening process of gold hydrometallurgy can refer to the paper [19]. To make the safe control decisions to remove the abnormities, we will establish the BN model to analyze the causes and the corresponding removing measures of abnormities. The related variables include three types: the safe control decisions ( $A^{\prime}$ : Change the opening degree of valve 3 and the power of slurry pump 1; $B^{\prime}$ : Change the opening degree of valve 4; $C^{\prime}$ : Change the opening degree of valve 1.), the abnormal conditions ( $D^{\prime}$ : The underflow rate is too low; $E^{\prime}$ : The underflow concentration is too high; $F^{\prime}$ : The buffer slot 1 under the thickener is empty) and the corresponding phenomena ( $G^{\prime}$ : The motor current in the thickener is too large; $H^{\prime}$ : The bed pressure in the thickener is too high; $I^{\prime}$ : The electricity of slurry pump 1 is not stable). To establish the BN structure, the relationships between the arbitrary two variables all need to be determined. The proposed method in the Section 4 will be applied to establish the BN structure for the thickening process of gold hydrometallurgy. In the following examples, we will take the relationships among the variables $A^{\prime}$ , $B^{\prime}$ and $D^{\prime}$ as the examples to verify the proposed method for the consistent and inconsistent information.

5.2 Consistent information

Example 5.1 Taking the relationships between the variables $A^{\prime}$ and $D^{\prime}$ as the example, the alternative relationships include that $A^{\prime}\to D^{\prime}$ ; $A^{\prime}\leftarrow D^{\prime}$ ; $A^{\prime}\neq D^{\prime}$ . We consider the condition that the information from the group experts is consistent. Four experts are invited to express their opinions on the relationship. To determine the relationship between $A^{\prime}$ and $D^{\prime}$ , the proposed scheme in the Section 4.2 is carried out.

Step 1.
Obtain the knowledge base from the group experts for every alternative relationship between the BN nodes in the intuitionistic fuzzy environment. The matrix is in the following form.

$\displaystyle D=\left({{\begin{array}[]{*{20}c}{\langle 0.68,0.22\rangle}&{% \langle 0.22,0.68\rangle}&{\langle 0.1,0.8\rangle}\\ {\langle 0.53,0.39\rangle}&{\langle 0.28,0.64\rangle}&{\langle 0.19,0.73% \rangle}\\ {\langle 0.45,0.5\rangle}&{\langle 0.35,0.6\rangle}&{\langle 0.2,0.75\rangle}% \\ {\langle 0.7,0.15\rangle}&{\langle 0.12,0.73\rangle}&{\langle 0.18,0.67\rangle% }\\ \end{array}}}\right)$

Table 1
The aggregation results for the relationship between the nodes $A^{\prime}$ and $D^{\prime}$

The aggregation way $A^{\prime}\to D^{\prime}$ $A^{\prime}\leftarrow D^{\prime}$ $A^{\prime}\neq D^{\prime}$ The relative closeness

coefficients

IFWA $<$ 0.6110, 0.2733 $>$ $<$ 0.2412, 0.6642 $>$ $<$ 0.1662, 0.7358 $>$ $\left[\begin{array}[]{c}0.6513\\ 0.3068\\ 0.2406\\ \end{array}\right]^{T}$

The aggregation operator from the D-S theory $<$ 0.5631, 0.3394 $>$ $<$ 0.2528, 0.6497 $>$ $<$ 0.1338, 0.7687 $>$ $\left[\begin{array}[]{c}0.6019\\ 0.3192\\ 0.2108\\ \end{array}\right]^{T}$

From the opinions of group experts, we can find that the four experts are all hold the higher membership degree on the relationship $A^{\prime}\to D^{\prime}$ . Therefore, the opinions from the group experts are consistent.
Step 2.
Calculate the weight of every expert.

Based on Eqs (5) and (7), $\omega_{K}=[0.2707\,\,0.2346\,\,0.2261\,\,0.2686]^{T}$ .
Step 3.
Aggregate the information from the group experts to obtain the comprehensive evaluation on the alternative relationships between the BN nodes.

The aggregation operator IFWA and the aggregation operator from the D-S theory are used to aggregate the information from the group experts respectively. The aggregation results are shown in the Table 1.
Step 4.
Calculate the relative closeness coefficient of every alternative relationship.

Based on Eqs (8) and (9), the relative closeness coefficients $\xi_{j}(1\leqslant j\leqslant n)$ are shown in the Table 1.
Step 5.
Rank all relative closeness coefficients and select the most suitable relationship between the BN nodes.

By observing the opinions of four experts intuitively, we can infer that the aggregation result is $A^{\prime}\to D^{\prime}$ . Based on the obtained relative closeness coefficients, we can obtain that the most suitable relationship between the BN nodes $A^{\prime}$ and $D^{\prime}$ is also $A^{\prime}\to D^{\prime}$ , which owns the highest value among all the relative closeness coefficients. Therefore, we can conclude that when the information is consistent, the proposed aggregation scheme is effective to obtain the suitable relationship between the BN nodes.
5.3 Inconsistent information

The aggregation way	$A^{\prime}\to D^{\prime}$	$A^{\prime}\leftarrow D^{\prime}$	$A^{\prime}\neq D^{\prime}$	The relative closeness
IFWA	$<$ 0.6110, 0.2733 $>$	$<$ 0.2412, 0.6642 $>$	$<$ 0.1662, 0.7358 $>$	$\left[\begin{array}[]{c}0.6513\\ 0.3068\\ 0.2406\\ \end{array}\right]^{T}$
The aggregation operator from the D-S theory	$<$ 0.5631, 0.3394 $>$	$<$ 0.2528, 0.6497 $>$	$<$ 0.1338, 0.7687 $>$	$\left[\begin{array}[]{c}0.6019\\ 0.3192\\ 0.2108\\ \end{array}\right]^{T}$

Example 5.2 Taking the relationships between the variables $B^{\prime}$ and $D^{\prime}$ as the example, the alternative relationships include that $B^{\prime}\to D^{\prime}$ ; $B^{\prime}\leftarrow D^{\prime}$ ; $B^{\prime}\neq D^{\prime}$ . We consider the condition that the information from the group experts is inconsistent. Four experts are invited to express their opinions on the relationships. To determine the relationship between $B^{\prime}$ and $D^{\prime}$ , the same steps with the Example 5.1 are carried out. The specific steps are omitted, and the main results are shown.

The obtained knowledge base matrix from the group experts for every alternative relationship between $B^{\prime}$ and $D^{\prime}$ in the intuitionistic fuzzy environment is shown in the following form.

$\displaystyle D=\left({{\begin{array}[]{*{20}c}{\langle 0,0.2\rangle}&{\langle 0% .9,0.05\rangle}&{\langle 0.1,0.2\rangle}\\ {\langle 0.6,0.3\rangle}&{\langle 0.25,0.65\rangle}&{\langle 0.15,0.75\rangle}% \\ {\langle 0.7,0.25\rangle}&{\langle 0.2,0.75\rangle}&{\langle 0.1,0.85\rangle}% \\ {\langle 0.65,0.15\rangle}&{\langle 0.26,0.5\rangle}&{\langle 0.09,0.7\rangle}% \\ \end{array}}}\right)$

From the opinions of group experts, we can find that the first expert holds the view $B^{\prime}\leftarrow D^{\prime}$ , but the other three experts hold the view $B^{\prime}\to D^{\prime}$ . The opinions from the group experts are inconsistent.

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for the relationship between the nodes $B^{\prime}$ and $D^{\prime}$ are shown in the Table 2.

Table 2
The aggregation results for the relationship between the nodes $B^{\prime}$ and $D^{\prime}$

The aggregation way	$B^{\prime}\to D^{\prime}$	$B^{\prime}\leftarrow D^{\prime}$	$B^{\prime}\neq D^{\prime}$	The relative closeness
				coefficients
IFWA	$<$ 0.5804, 0.2211 $>$	$<$ 0.4732, 0.3966 $>$	$<$ 0.1110, 0.6005 $>$	$\left[{{\begin{array}[]{*{20}c}{{0.6499}}\\ {{0.5339}}\\ {{0.3100}}\\ \end{array}}}\right]^{T}$
The aggregation operator from the D-S theory	$<$ 0.5166, 0.2446 $>$	$<$ 0.3528, 0.5357 $>$	$<$ 0.1022, 0.6748 $>$	$\left[{{\begin{array}[]{*{20}c}{{0.6098}}\\ {{0.4177}}\\ {{0.2659}}\\ \end{array}}}\right]^{T}$

By observing the opinions of four experts intuitively, we can infer that the aggregation result is $B^{\prime}\to D^{\prime}$ . Based on the obtained relative closeness coefficients, we can obtain that the most suitable relationship between the BN nodes $B^{\prime}$ and $D^{\prime}$ is $B^{\prime}\to D^{\prime}$ , which owns the highest value among all the relative closeness coefficients. Therefore, we can conclude that when the information is inconsistent, the proposed aggregation scheme is effective to make decision.

From the obtained knowledge base matrix in the Example 5.2, we can find that no membership degree is equal to one and no non-membership degree is equal to zero. Therefore, the Examples 5.3 and 5.4 will consider the condition that the knowledge base matrix includes the special membership degree and non-membership degree.

Example 5.3 Taking the relationships between two nodes $A^{\prime}$ and $B^{\prime}$ as the example, the alternative relationships include that $A^{\prime}\to B^{\prime}$ ; $A^{\prime}\leftarrow B^{\prime}$ ; $A^{\prime}\neq B^{\prime}$ . Four experts are invited to express their opinions on the relationship. To determine the relationship between $A^{\prime}$ and $B^{\prime}$ , the same steps with the Example 5.1 are carried out. The specific steps are omitted, and the main results are shown.

The obtained knowledge base from the group experts for every alternative relationship between $A^{\prime}$ and $B^{\prime}$ in the intuitionistic fuzzy environment is shown in the following form.

$\displaystyle D=\left({{\begin{array}[]{*{20}c}{\langle 0.9,0.05\rangle}&{% \langle 0.01,0.98\rangle}&{\langle 0.09,0.71\rangle}\\ {\langle 0,0.99\rangle}&{\langle 0.01,0.98\rangle}&{\langle 0.99,0\rangle}\\ {\langle 0.02,0.97\rangle}&{\langle 0.03,0.96\rangle}&{\langle 0.95,0.04% \rangle}\\ {\langle 0.1,0.89\rangle}&{\langle 0.9,0.02\rangle}&{\langle 0,0.8\rangle}\\ \end{array}}}\right)$

From the opinions of group experts, we can find that the first expert holds the view $A^{\prime}\to B^{\prime}$ , the second and the third experts hold the view $A^{\prime}\neq B^{\prime}$ , and the fourth expert holds the view $A^{\prime}\leftarrow B^{\prime}$ . The opinions from the group experts are inconsistent, and the opinions from the second expert include the condition that the non-membership degree is equal to zero.

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for the relationship between the nodes $A^{\prime}$ and $B^{\prime}$ are shown in the Table 3.

Table 3

The aggregation results for the relationship between the nodes $A^{\prime}$ and $B^{\prime}$ including $\nu=0$

The aggregation way	$A^{\prime}\to B^{\prime}$	$A^{\prime}\leftarrow B^{\prime}$	$A^{\prime}\neq B^{\prime}$	The relative closeness
				coefficients
IFWA	$<$ 0.4392, 0.4718 $>$	$<$ 0.4328, 0.3820 $>$	$<$ 0.8660, 0 $>$	$\left[{{\begin{array}[]{*{20}c}{{0.4851}}\\ {{0.5214}}\\ {{0.8818}}\\ \end{array}}}\right]^{T}$
The aggregation operator from the D-S theory	$<$ 0.2498, 0.7307 $>$	$<$ 0.0122, 0.9609 $>$	$<$ 0.7099, 0.1891 $>$	$\left[{{\begin{array}[]{*{20}c}{{0.2641}}\\ {{0.0380}}\\ {{0.7365}}\\ \end{array}}}\right]^{T}$

By observing the opinions of group experts intuitively, we can infer that the aggregation result is $A^{\prime}\neq B^{\prime}$ . Based on the obtained relative closeness coefficients, we can obtain that the most suitable relationship between the nodes $A^{\prime}$ and $B^{\prime}$ is $A^{\prime}\neq B^{\prime}$ , which owns the highest value among all the relative closeness coefficients. Therefore, we can conclude that when the information is inconsistent, which includes $\nu=0$ , the proposed aggregation scheme is effective to make decision. However, there is a problem in the results based on the aggregation operator IFWA shown in the Table 3. Because the condition that the non-membership degree is equal to zero exists, it covers the influence of other non-membership degrees, and the non-membership degree of final aggregation result is equal to zero for the relationship $A^{\prime}\neq B^{\prime}$ . By analyzing the IFWA operator Eq. (2), we can obtain that if one non-membership degree is equal to zero, the non-membership degree of aggregation result is equal to zero, which is the drawback of IFWA operator. For the same condition, based on the aggregation operator from the D-S theory, the aggregation result is not covered by the special value of non-membership degree. This operator overcomes the drawback of IFWA operator.

Example 5.4 Based on the Example 5.3, we consider the condition that the membership degree is equal to one further. The opinion of first expert for $A^{\prime}\to B^{\prime}$ is replaced by $\langle 1,0\rangle$ , and other IFVs are the same with the Example 5.3. The obtained knowledge base from the group experts for every alternative relationship between $A^{\prime}$ and $B^{\prime}$ in the intuitionistic fuzzy environment is shown in the following form.

$\displaystyle D=\left({{\begin{array}[]{*{20}c}{\langle 1,0\rangle}&{\langle 0% .01,0.98\rangle}&{\langle 0.09,0.71\rangle}\\ {\langle 0,0.99\rangle}&{\langle 0.01,0.98\rangle}&{\langle 0.99,0\rangle}\\ {\langle 0.02,0.97\rangle}&{\langle 0.03,0.96\rangle}&{\langle 0.95,0.04% \rangle}\\ {\langle 0.1,0.89\rangle}&{\langle 0.9,0.02\rangle}&{\langle 0,0.8\rangle}\\ \end{array}}}\right)$

For the relationship between the nodes $A^{\prime}$ and $B^{\prime}$ , the aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory are shown in the Table 4. Based on the analysis in the Example 5.3, we can know that the most suitable relationship between the nodes $A^{\prime}$ and $B^{\prime}$ should be $A^{\prime}\neq B^{\prime}$ . The aggregation results based on the aggregation operator from the D-S theory can obtain the right conclusion. However, the aggregation result based on the aggregation operator IFWA is $A^{\prime}\to B^{\prime}$ . By analyzing the IFWA operator Eq. (2), we can obtain that if one membership degree is equal to one, it will cover the influence of other membership degrees. The aggregation result of all the membership degrees is equal to one, which is the drawback of IFWA operator. For the same condition, based on the aggregation operator from the D-S theory, the aggregation result is not covered by the special membership degree. This operator overcomes the drawback of IFWA operator.

In addition, we can also use another way to overcome this drawback of IFWA operator in some degree. The membership degree which is equal to one can be replaced by the membership degree which is close to one. For example, the opinion of first expert is replaced by not $\langle 1,0\rangle$ but $\langle 0.99,0\rangle$ , the aggregation results are shown in the fourth row in the Table 4. The aggregation result changes from $A^{\prime}\to B^{\prime}$ to $A^{\prime}\neq B^{\prime}$ . Therefore, this way can overcome the drawback of IFWA operator in some degree.

Table 4

The aggregation results for the relationship between the nodes $A^{\prime}$ and $B^{\prime}$ including $\nu=0$ and $\mu=1$

The aggregation way	$A^{\prime}\to B^{\prime}$	$A^{\prime}\leftarrow B^{\prime}$	$A^{\prime}\neq B^{\prime}$	The relative closeness
				coefficients
IFWA	$<$ 1, 0 $>$	$<$ 0.4306, 0.3846 $>$	$<$ 0.8642, 0 $>$	$\left[{{\begin{array}[]{*{20}c}1\\ {{0.5194}}\\ {{0.8804}}\\ \end{array}}}\right]^{T}$
The aggregation operator from the D-S theory	$<$ 0.2904, 0.7020 $>$	$<$ 0.0121, 0.9611 $>$	$<$ 0.7043, 0.1941 $>$	$\left[{{\begin{array}[]{*{20}c}{{0.2957}}\\ {{0.0379}}\\ {{0.7316}}\\ \end{array}}}\right]^{T}$
IFWA ( $\mu=1$ is replaced by $\mu=0.99$ )	$<$ 0.6830, 0 $>$	$<$ 0.4308, 0.3844 $>$	$<$ 0.8643, 0 $>$	$\left[{{\begin{array}[]{*{20}c}{{0.7593}}\\ {{0.5196}}\\ {{0.8805}}\\ \end{array}}}\right]^{T}$

Based on the aggregation results of Examples 5.1–5.4, in the most conditions, two aggregation operators are all effective to make decision. But when the knowledge base matrix includes special membership degree or non-membership degree, the aggregation operator from the D-S theory can overcome some drawbacks of IFWA operator, which is the better choice to make decision than the IFWA aggregation operator.

Based on the aggregation results of Examples 5.1–5.4, the relationships among the nodes $A^{\prime}$ , $B^{\prime}$ and $D^{\prime}$ can be described. The relationships between other variables can be determined by the same way. Finally, based on the obtained the relationships among all the variables, the BN structure can be established for the thickening process of gold hydrometallurgy, which is shown in Fig. 3.

To show the role of hesitancy degree in the decision-making process for the inconsistent information, in the following Examples 5.5 and 5.6, we will observe the change of aggregation results by changing the value of hesitancy degree in the IFVs.

Example 5.5 Taking the relationships between two nodes $Q$ and $P$ as the example, the alternative relationships include that $Q\to P$ ; $Q\leftarrow P$ ; $Q\neq P$ . Three experts are invited to express their opinions on every alternative relationship.

The obtained knowledge base matrix from the group experts for every alternative relationship between $Q$ and $P$ in the intuitionistic fuzzy environment is shown in the following form.

$\displaystyle D_{1}=\left({{\begin{array}[]{*{20}c}{\langle 0,1\rangle}&{% \langle 0.9,0.1\rangle}&{\langle 0.1,0.9\rangle}\\ {\langle 0.6,0.4\rangle}&{\langle 0.25,0.75\rangle}&{\langle 0.15,0.85\rangle}% \\ {\langle 0.7,0.3\rangle}&{\langle 0.2,0.8\rangle}&{\langle 0.1,0.9\rangle}\\ \end{array}}}\right)$

In $D_{1}$ , the hesitancy degrees of three experts are all set as zero. From the opinions of group experts, we can find that the first expert holds the view $Q\leftarrow P$ , but other two experts hold the view $Q\to P$ . The opinions from the group experts are inconsistent. By observing the opinions of three experts intuitively, we can find that the opinion of first expert on $Q\to P$ is negative completely, and the membership degree of $Q\leftarrow P$ is very big. Therefore, we can infer that the first expert plays the most important role among all the experts, and the aggregation result is $Q\leftarrow P$ .

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for $D_{1}$ are shown in the Table 5. From the Table 5, we can get that the aggregation results based on the aggregation operators are the same with the inferred result.

Table 5

The aggregation results for $D_{1}$

The aggregation way	The relative closeness coefficients	The relationship between the
		nodes $Q$ and $P$
IFWA	$\left[{{\begin{array}[]{*{20}c}{{0.4846}}\\ {{0.6427}}\\ {{0.1152}}\\ \end{array}}}\right]^{T}$	$Q\leftarrow P$
The aggregation operator from the D-S theory	$\left[{{\begin{array}[]{*{20}c}{{0.3695}}\\ {{0.4839}}\\ {{0.0830}}\\ \end{array}}}\right]^{T}$	$Q\leftarrow P$

Figure 3.

The established BN structure by the proposed method for the thickening process of gold hydrometallurgy.

At first, to test the role of hesitancy degree, we increase the hesitancy degree of first expert for $D_{1}$ , and the hesitancy degrees of other two experts keep unchanged. The matrix $D_{1}$ is transformed into the matrix $D_{2}$ .

$\displaystyle D_{2}=\left({{\begin{array}[]{*{20}c}{\langle 0,0.2\rangle}&{% \langle 0.9,0.05\rangle}&{\langle 0.1,0.2\rangle}\\ {\langle 0.6,0.4\rangle}&{\langle 0.25,0.75\rangle}&{\langle 0.15,0.85\rangle}% \\ {\langle 0.7,0.3\rangle}&{\langle 0.2,0.8\rangle}&{\langle 0.1,0.9\rangle}\\ \end{array}}}\right)$

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for $D_{2}$ are shown in the Table 6.

Table 6

The aggregation results for $D_{2}$

The aggregation way	The relative closeness coefficients	The relationship between the
		nodes $Q$ and $P$
IFWA	$\left[{{\begin{array}[]{*{20}c}{0.6112}\\ {0.5558}\\ {0.3033}\\ \end{array}}}\right]^{T}$	$Q\to P$
The aggregation operator from the D-S theory	$\left[{{\begin{array}[]{*{20}c}{0.5942}\\ {0.3522}\\ {0.2225}\\ \end{array}}}\right]^{T}$	$Q\to P$

Furthermore, for $D_{2}$ , we increase the hesitancy degree of first expert to the biggest value, and the hesitancy degrees of other two experts keep unchanged. The matrix $D_{2}$ is transformed into the matrix $D_{3}$ .

$\displaystyle D_{3}=\left({{\begin{array}[]{*{20}c}{\langle 0,0\rangle}&{% \langle 0.9,0\rangle}&{\langle 0.1,0\rangle}\\ {\langle 0.6,0.4\rangle}&{\langle 0.25,0.75\rangle}&{\langle 0.15,0.85\rangle}% \\ {\langle 0.7,0.3\rangle}&{\langle 0.2,0.8\rangle}&{\langle 0.1,0.9\rangle}\\ \end{array}}}\right)$

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for $D_{3}$ are shown in the Table 7.

Table 7

The aggregation results for $D_{3}$

The aggregation way	The relative closeness coefficients	The relationship between the
		nodes $Q$ and $P$
IFWA	$\left[{{\begin{array}[]{*{20}c}{0.7031}\\ {0.6562}\\ {0.5318}\\ \end{array}}}\right]^{T}$	$Q\to P$
The aggregation operator from the D-S theory	$\left[{{\begin{array}[]{*{20}c}{0.6172}\\ {0.3186}\\ {0.2270}\\ \end{array}}}\right]^{T}$	$Q\to P$

Comparing the aggregation results of $D_{1}D_{2}$ and $D_{3}$ in the Tables 5–7, we can find that as the increase of hesitancy degree of first expert, the aggregation results change from $Q\leftarrow P$ to $Q\to P$ .

Based on the aggregation results of Example 5.5, we can conclude that for the aggregation of inconsistent information, the aggregation results change as the change of hesitancy degrees. The hesitancy degree plays the important role in the decision-making process. The expert who owns bigger membership degree and less hesitancy degree plays more important role in the aggregation results.

In the Example 5.5, two of three experts own the consistent opinions. In the following example, we will consider the extreme condition that no one expert holds the same opinions, that is to say, the opinions of group experts are conflict.

Example 5.6 Taking the relationships between two nodes $Q$ and $P$ as the example, the alternative relationships include that $Q\to P$ ; $Q\leftarrow P$ ; $Q\neq P$ . Three experts are invited to express their opinions on the relationships.

The obtained knowledge base matrix from the group experts for every alternative relationship between $Q$ and $P$ in the intuitionistic fuzzy environment is shown in the following form.

$\displaystyle D_{4}=\left({{\begin{array}[]{*{20}c}{\langle 0.9,0.1\rangle}&{% \langle 0.01,0.99\rangle}&{\langle 0.1,0.9\rangle}\\ {\langle 0.01,0.99\rangle}&{\langle 0.1,0.9\rangle}&{\langle 0.9,0.1\rangle}\\ {\langle 0.1,0.9\rangle}&{\langle 0.9,0.1\rangle}&{\langle 0.01,0.99\rangle}\\ \end{array}}}\right)$

In $D_{4}$ , the hesitancy degrees of three experts are all set as zero. From the opinions of group experts, we can find that the first expert holds the view $Q\to P$ , the second expert holds the view $Q\neq P$ , and the third expert holds the view $Q\leftarrow P$ . The opinions from the group experts are inconsistent absolutely. At the same time, the matrix $D_{4}$ is symmetrical, and the knowledge measures of group experts are the same based on Eq. (5). By observing the opinions of three experts intuitively, we cannot obtain the decision result.

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for $D_{4}$ are shown in the Table 8. From the Table 8, the relative closeness coefficients of all the experts are the same values, we cannot obtain the aggregation results, which is the same with the inferred conclusion.

Table 8

The aggregation results for $D_{4}$

The aggregation way	The relative closeness coefficients	The relationship between the
		nodes $Q$ and $P$
IFWA	$\left[{{\begin{array}[]{*{20}c}{{0.3333}}\\ {{0.3333}}\\ {{0.3333}}\\ \end{array}}}\right]^{T}$	No results are obtained
The aggregation operator from the D-S theory	$\left[{{\begin{array}[]{*{20}c}{{0.3333}}\\ {{0.3333}}\\ {{0.3333}}\\ \end{array}}}\right]^{T}$	No results are obtained

At first, to test the role of hesitancy degree, we increase the hesitancy degree of second expert for $D_{4}$ . The hesitancy degrees of other two experts keep unchanged. The matrix $D_{4}$ is transformed into the matrix $D_{5}$ .

$\displaystyle D_{5}=\left({{\begin{array}[]{*{20}c}{\langle 0.9,0.1\rangle}&{% \langle 0.01,0.99\rangle}&{\langle 0.1,0.9\rangle}\\ {\langle 0.01,0.5\rangle}&{\langle 0.1,0.5\rangle}&{\langle 0.9,0.01\rangle}\\ {\langle 0.1,0.9\rangle}&{\langle 0.9,0.1\rangle}&{\langle 0.01,0.99\rangle}\\ \end{array}}}\right)$

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for $D_{5}$ are shown in the Table 9.

Comparing the aggregation results of $D_{4}$ and $D_{5}$ in the Tables 8 and 9, we can find that as the increase of hesitancy degree of second expert, the aggregation results change from no results to $Q\to P$ . We can find that $Q\to P$ is the opinion of first expert, and the aggregation results can be changed by changing the value of hesitancy degree. Therefore, we infer that if the hesitancy degree of first expert is increased, the aggregation results will continue to change. To verify this conclusion, for $D_{5}$ , we increase the hesitancy degree of first expert, and the hesitancy degrees of other two experts keep unchanged. The matrix $D_{5}$ is transformed into the matrix $D_{6}$ .

$\displaystyle D_{6}=\left({{\begin{array}[]{*{20}c}{\langle 0.9,0.01\rangle}&{% \langle 0.01,0.5\rangle}&{\langle 0.1,0.5\rangle}\\ {\langle 0.01,0.5\rangle}&{\langle 0.1,0.5\rangle}&{\langle 0.9,0.01\rangle}\\ {\langle 0.1,0.9\rangle}&{\langle 0.9,0.1\rangle}&{\langle 0.01,0.99\rangle}\\ \end{array}}}\right)$

The aggregation results based on the aggregation operator IFWA and the aggregation operator from the D-S theory for $D_{6}$ are shown in the Table 10.

Comparing the aggregation results of $D_{5}$ and $D_{6}$ in the Tables 9 and 10, we can find that as the increase of hesitancy degree of first expert, the aggregation results change from $Q\to P$ to $Q\leftarrow P$ . We can find that $Q\leftarrow P$ is the opinion of third expert.

Table 9

The aggregation results for $D_{5}$

The aggregation way	The relative closeness coefficients	The relationship between the
		nodes $Q$ and $P$
IFWA	$\left[{{\begin{array}[]{*{20}c}{{0.6104}}\\ {{0.6040}}\\ {{0.5911}}\\ \end{array}}}\right]^{T}$	$Q\to P$
The aggregation operator from the D-S theory	$\left[{{\begin{array}[]{*{20}c}{{0.4339}}\\ {{0.4049}}\\ {{0.2472}}\\ \end{array}}}\right]^{T}$	$Q\to P$

Table 10

The aggregation results for $D_{6}$

The aggregation way	The relative closeness coefficients	The relationship between the
		nodes $Q$ and $P$
IFWA	$\left[{{\begin{array}[]{*{20}c}{{0.6288}}\\ {{0.6538}}\\ {{0.6246}}\\ \end{array}}}\right]^{T}$	$Q\leftarrow P$
The aggregation operator from the D-S theory	$\left[{{\begin{array}[]{*{20}c}{{0.3941}}\\ {{0.5412}}\\ {{0.3612}}\\ \end{array}}}\right]^{T}$	$Q\leftarrow P$

By the above aggregation results, we can find that for the aggregation of conflict information, when the matrix is symmetrical, and the knowledge measures of group experts are the same, that is to say, when the group opinions are inconsistent entirely, the aggregation results cannot be obtained. However, the aggregation result changes as the change of hesitancy degree, which tends to the opinion of expert who owns the less hesitancy degree. Therefore, based on the simulation analysis in the Examples 5.2–5.6, we can conclude that the expert who owns bigger membership degree and less hesitancy degree plays the most important role in the aggregation of inconsistent information.

6. Conclusions

In this paper, the relationship determination problem between the BN nodes is transformed into the GDM problem under the intuitionistic fuzzy environment. The scheme of GDM based on the IFS is proposed. The expert knowledge about the relationships between the nodes is collected in the form of IFVs. When collecting the expert knowledge, the subjective and the objective information are integrated. For the consistent and inconsistent information, the IFWA operator and the aggregation operator from the D-S theory are applied to aggregate the opinions from the group experts respectively. The proposed scheme is applied to solve the practical problem for the thickening process of gold hydrometallurgy. The simulation results imply that when the knowledge base matrix includes the special membership degree or non-membership degree, the aggregation operator from the D-S theory can overcome some drawbacks of IFWA operator, which is the better choice to make decision. Furthermore, the role of hesitancy degree is analyzed by changing the value of hesitancy degree. We can conclude that the expert who owns bigger membership degree and less hesitancy degree plays the most important role in the aggregation results.

Footnotes

Acknowledgments

This work was supported by the National Nature Science Foundation of China [grant numbers 61533007; 61873049], the National Key Research and Development Program of China [grant numbers 2017YFB0304205].

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